One mole of an ideal gas undergoes two different cyclic processes I and II, as shown in the 𝑃-𝑉 diagrams below. In cycle I, processes 𝑎, 𝑏, 𝑐 and 𝑑 are isobaric, isothermal, isobaric and isochoric, respectively. In cycle II, processes 𝑎 ′ , 𝑏 ′ , 𝑐 ′ and 𝑑 ′ are isothermal, isochoric, isobaric and isochoric, respectively. The total work done during cycle I is 𝑊𝐼 and that during cycle II is 𝑊𝐼𝐼. The ratio 𝑊𝐼/𝑊𝐼𝐼 is _______.

 To determine the ratio 

$\frac{W_I}{W_{II}}$ of the work done during cycle I to that during cycle II, we need to analyze each cycle step-by-step and calculate the work done in each process.

Cycle I:

1. Process $a$ (Isobaric Expansion):

   • Work done: $W_a=P_1(V_2-V_1)$

2. Process $b$ (Isothermal Compression):

   • Work done: $W_b=nR{T}_2\ln \left(\frac{V_1}{V_2}\right)$

3. Process $c$ (Isobaric Compression):

   • Work done: $W_c=P_2(V_1-V_2)$

4. Process $d$ (Isochoric Process):

   • Work done: $W_d=0$ (since volume is constant)

Total Work for Cycle I:

$$W_I=W_a+W_b+W_c+W_d$$$$W_I=P_1(V_2-V_1)+nR{T}_2\ln \left(\frac{V_1}{V_2}\right)+P_2(V_1-V_2)$$

Cycle II:

1. Process $a^{\mathrm{\prime }}$ (Isothermal Expansion):

   • Work done: $W_{a^{\mathrm{\prime }}}=nR{T}_1\ln \left(\frac{V_2}{V_1}\right)$

2. Process $b^{\mathrm{\prime }}$ (Isochoric Process):

   • Work done: $W_{b^{\mathrm{\prime }}}=0$ (since volume is constant)

3. Process $c^{\mathrm{\prime }}$ (Isobaric Compression):

   • Work done: $W_{c^{\mathrm{\prime }}}=P_2(V_1-V_2)$

4. Process $d^{\mathrm{\prime }}$ (Isochoric Process):

   • Work done: $W_{d^{\mathrm{\prime }}}=0$ (since volume is constant)

Total Work for Cycle II:

$$W_{II}=W_{a^{\mathrm{\prime }}}+W_{b^{\mathrm{\prime }}}+W_{c^{\mathrm{\prime }}}+W_{d^{\mathrm{\prime }}}$$$$W_{II}=nR{T}_1\ln \left(\frac{V_2}{V_1}\right)+P_2(V_1-V_2)$$

Ratio $\frac{W_I}{W_{II}}$:

$$\frac{W_I}{W_{II}}=\frac{P_1(V_2-V_1)+nR{T}_2\ln \left(\frac{V_1}{V_2}\right)+P_2(V_1-V_2)}{nR{T}_1\ln \left(\frac{V_2}{V_1}\right)+P_2(V_1-V_2)}$$

This ratio depends on the specific values of $P_1,P_2,V_1,V_2,T_1,$ and $T_2$. Without numerical values, we cannot simplify this expression further. However, the ratio $\frac{W_I}{W_{II}}$ can be calculated if the specific values of these parameters are provided.